Average Error: 6.5 → 2.3
Time: 14.5s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(1 + z \cdot z\right) \cdot y = -\infty:\\ \;\;\;\;\frac{1}{y} \cdot \left(\frac{\frac{\frac{1}{x}}{z}}{z} - \frac{\frac{\frac{1}{x}}{z \cdot z}}{z \cdot z} \cdot 1\right)\\ \mathbf{elif}\;\left(1 + z \cdot z\right) \cdot y \le 1.862479689839605283071532494157124524407 \cdot 10^{307}:\\ \;\;\;\;\frac{1}{\left(\left(1 + z \cdot z\right) \cdot y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(\frac{\frac{\frac{1}{x}}{z}}{z} - \frac{\frac{\frac{1}{x}}{z \cdot z}}{z \cdot z} \cdot 1\right)\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\left(1 + z \cdot z\right) \cdot y = -\infty:\\
\;\;\;\;\frac{1}{y} \cdot \left(\frac{\frac{\frac{1}{x}}{z}}{z} - \frac{\frac{\frac{1}{x}}{z \cdot z}}{z \cdot z} \cdot 1\right)\\

\mathbf{elif}\;\left(1 + z \cdot z\right) \cdot y \le 1.862479689839605283071532494157124524407 \cdot 10^{307}:\\
\;\;\;\;\frac{1}{\left(\left(1 + z \cdot z\right) \cdot y\right) \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{y} \cdot \left(\frac{\frac{\frac{1}{x}}{z}}{z} - \frac{\frac{\frac{1}{x}}{z \cdot z}}{z \cdot z} \cdot 1\right)\\

\end{array}
double f(double x, double y, double z) {
        double r18567576 = 1.0;
        double r18567577 = x;
        double r18567578 = r18567576 / r18567577;
        double r18567579 = y;
        double r18567580 = z;
        double r18567581 = r18567580 * r18567580;
        double r18567582 = r18567576 + r18567581;
        double r18567583 = r18567579 * r18567582;
        double r18567584 = r18567578 / r18567583;
        return r18567584;
}
↓
double f(double x, double y, double z) {
        double r18567585 = 1.0;
        double r18567586 = z;
        double r18567587 = r18567586 * r18567586;
        double r18567588 = r18567585 + r18567587;
        double r18567589 = y;
        double r18567590 = r18567588 * r18567589;
        double r18567591 = -inf.0;
        bool r18567592 = r18567590 <= r18567591;
        double r18567593 = r18567585 / r18567589;
        double r18567594 = 1.0;
        double r18567595 = x;
        double r18567596 = r18567594 / r18567595;
        double r18567597 = r18567596 / r18567586;
        double r18567598 = r18567597 / r18567586;
        double r18567599 = r18567596 / r18567587;
        double r18567600 = r18567599 / r18567587;
        double r18567601 = r18567600 * r18567585;
        double r18567602 = r18567598 - r18567601;
        double r18567603 = r18567593 * r18567602;
        double r18567604 = 1.8624796898396053e+307;
        bool r18567605 = r18567590 <= r18567604;
        double r18567606 = r18567590 * r18567595;
        double r18567607 = r18567585 / r18567606;
        double r18567608 = r18567605 ? r18567607 : r18567603;
        double r18567609 = r18567592 ? r18567603 : r18567608;
        return r18567609;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.5
Target5.7
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (+ 1.0 (* z z))) < -inf.0 or 1.8624796898396053e+307 < (* y (+ 1.0 (* z z)))

    1. Initial program 18.6

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied div-inv18.6

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{x}}}{y \cdot \left(1 + z \cdot z\right)}\]

    4. Applied times-frac14.3

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}}\]

    5. Taylor expanded around inf 14.4

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\left(\frac{1}{x \cdot {z}^{2}} - 1 \cdot \frac{1}{x \cdot {z}^{4}}\right)}\]

    6. Simplified5.7

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\left(\frac{\frac{\frac{1}{x}}{z}}{z} - 1 \cdot \frac{\frac{\frac{1}{x}}{z \cdot z}}{z \cdot z}\right)}\]

    if -inf.0 < (* y (+ 1.0 (* z z))) < 1.8624796898396053e+307

    1. Initial program 0.3

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied div-inv0.3

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{x}}}{y \cdot \left(1 + z \cdot z\right)}\]

    4. Applied associate-/l*0.6

      \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot \left(1 + z \cdot z\right)}{\frac{1}{x}}}}\]

    5. Simplified0.6

      \[\leadsto \frac{1}{\color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z \cdot z\right) \cdot y = -\infty:\\ \;\;\;\;\frac{1}{y} \cdot \left(\frac{\frac{\frac{1}{x}}{z}}{z} - \frac{\frac{\frac{1}{x}}{z \cdot z}}{z \cdot z} \cdot 1\right)\\ \mathbf{elif}\;\left(1 + z \cdot z\right) \cdot y \le 1.862479689839605283071532494157124524407 \cdot 10^{307}:\\ \;\;\;\;\frac{1}{\left(\left(1 + z \cdot z\right) \cdot y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(\frac{\frac{\frac{1}{x}}{z}}{z} - \frac{\frac{\frac{1}{x}}{z \cdot z}}{z \cdot z} \cdot 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))